Optimal. Leaf size=70 \[ \frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \sec (a+b x)}}{3 b}+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{3/2}}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3771, 2641} \[ \frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \sec (a+b x)}}{3 b}+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int (c \sec (a+b x))^{5/2} \, dx &=\frac {2 c (c \sec (a+b x))^{3/2} \sin (a+b x)}{3 b}+\frac {1}{3} c^2 \int \sqrt {c \sec (a+b x)} \, dx\\ &=\frac {2 c (c \sec (a+b x))^{3/2} \sin (a+b x)}{3 b}+\frac {1}{3} \left (c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx\\ &=\frac {2 c^2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \sec (a+b x)}}{3 b}+\frac {2 c (c \sec (a+b x))^{3/2} \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 51, normalized size = 0.73 \[ \frac {2 c^2 \sqrt {c \sec (a+b x)} \left (\tan (a+b x)+\sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \sec \left (b x + a\right )} c^{2} \sec \left (b x + a\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 128, normalized size = 1.83 \[ -\frac {2 \left (-1+\cos \left (b x +a \right )\right ) \left (i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right )-\cos \left (b x +a \right )+1\right ) \cos \left (b x +a \right ) \left (\cos \left (b x +a \right )+1\right )^{2} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}}}{3 b \sin \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sec {\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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